Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1186 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,13)(3,8,11,7,12,16,14,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,31,32,29)(26,30);(1,3)(2,7)(4,14)(5,11)(6,12)(8,13)(9,16)(10,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31);(1,3)(2,8)(4,14)(5,11)(6,12)(7,9)(10,16)(13,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72261029, #72261063 |
| maximal normal subgroup of | #72261029, #72261063 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22846);